Generalized (q,w)-Euler Numbers and Polynomials Associated with p-Adic q-Integral on ℤp

Recently, many mathematicians have studied in the area of the Euler numbers and polynomials see 1–15 . The Euler numbers and polynomials possess many interesting properties and arising in many areas of mathematics and physics. In 14 , we introduced that Euler equation En x 0 has symmetrical roots for x 1/2 see 14 . It is the aim of this paper to observe an interesting phenomenon of “scattering” of the zeros of the generalized q,w -Euler polynomials En,q,w x : a in complex plane. Throughout this paper, we use the following notations. By Zp, we denote the ring of p-adic rational integers, Qp denotes the field of p-adic rational numbers, Cp denotes the completion of algebraic closure of Qp, N denotes the set of natural numbers, Z denotes the ring of rational integers, Q denotes the field of rational numbers, C denotes the set of complex numbers, and Z N ∪ {0}. Let νp be the normalized exponential valuation of Cp with |p|p p−νp p p−1. When one talks of qextension, q is considered in many ways such as an indeterminate, a complex number q ∈ C, or p-adic number q ∈ Cp. If q ∈ C one normally assume that |q| < 1. If q ∈ Cp, we normally assume that |q − 1|p < p−1/ p−1 so that q exp x log q for |x|p ≤ 1